1. Field of the Invention
The invention relates in general to Mirau interferometry for optical testing and, in particular, to a modified Mirau interferometer producing orthogonally polarized beams suitable for achromatic phase-shifting interference microscopy.
2. Description of the Prior Art
Because of its simple configuration and corresponding relative ease of calibration, the Mirau interference objective is used widely to study the structure of surfaces requiring observation at higher magnification, typically 10× to 75×. It features a very compact interferometer incorporated in a single microscope objective, as illustrated schematically in FIG. 1. Light from an illuminator (not shown) is passed through a microscope objective 10 onto a beamsplitter consisting of a beamsplitting coating 12 between two identical transmissive plates 14 and 14′. The transmitted beam T proceeds to the test surface 16 of a sample object S, while the reflected beam R is directed to the reference surface 18, typically an aluminized spot on the front surface 20 of the microscope objective 10. The two beams T and R reflected from the test and reference surfaces, respectively, are recombined at the beamsplitter 12 and return through the objective 10 toward a detector element (not shown).
In an alternative Mirau embodiment, shown in FIG. 1B, the two plates 14,14′ are separated and the top plate 14 is positioned away from the coated beamsplitting plate 14′ in contact with the reference surface 18. As one skilled in the art would readily appreciate, the optical effect is the same because in both configurations the test and reference beams (T and R, respectively) traverse the same optical distance (due to the fact that the plates 14 and 14′ are identical).
The interference pattern formed in the image plane contours the deviations from flatness of the test surface 16. As is well understood in the art, if the beamsplitter in the Mirau interferometer (or the whole microscope objective incorporating the beamsplitter, or the object, or the reference mirror) is mounted on a piezoelectric translator (PZT), so that the optical path difference (OPD) can be changed by known amounts, very accurate measurements of surface profiles can be made using phase-shifting techniques. Mirau interferometers also provide inherent compensation for chromatic dispersion because the beamsplitting element 12 is contained within the two identical plates 14,14′. Thus, each beam (T and R) traverses the exact same optical length twice in each direction upon impinging on the beamsplitter, thereby negating any dispersive effects of the plates medium.
However, the performance of Mirau interferometers (as well as all other scanning interferometers) is affected by the fact that the algorithms used in optical scanning interferometry assume that the amplitude of interference signal modulation remains constant during the scan. In fact, that is almost never the case. In practice, the light intensity detected as a result of interference of the test and reference beams, which would be perfectly sinusoidal under ideal single-wavelength and zero-numerical-aperture conditions, as shown in FIG. 2, exhibits a modulation variation that affects the interferometric result even when narrow-band light is used (or non zero numerical aperture or both), as illustrated in FIG. 3.
Furthermore, it is well known that the use of monochromatic light is accompanied by the so-called 2π phase ambiguities that arise when the measurement range involves a change in the optical path difference (OPD) greater than a wavelength. Phase unwrapping techniques are used in the art, but they are effective only with smooth continuous surfaces and break down when the test surface exhibits a sharp step or a discontinuity.
One way of overcoming the 2π-ambiguity problem is by using white light and scanning the object along the height (z) axis. The position along the z axis yielding maximum visibility of the fringes (the coherence peak) for each pixel in the image is known to correspond to the height of the object at that point. The visibility peak can be located by shifting the phase of the reference wave by three or more known amounts at each step along the z axis and recording the corresponding values of intensity. These intensity values can then be used in conventional algorithms to evaluate the fringe visibility at that step. However, if the phase shifts are introduced by changes in the OPD, as is the case in conventional scanning interferometry, the value of the resulting phase shift varies inversely with the wavelength, thereby producing fringes with varying modulation, as shown in FIG. 3, and the calculated phase and modulation may contain errors.
This problem may be overcome by using a different technique of phase shifting involving a cycle of changes in the polarization of the light to produce the same phase shift, measured in degrees or radians, for all wavelengths. This phase shift, known in the art as the Pancharatnam phase shift (see S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci., A 41, 137-144, 1955), is a manifestation of the geometric phase and it can be used to generate any required wavelength-independent phase shift without changing the optical path difference. As a result, geometric phase-shifting has found many applications in interferometry.
In white-light interference, a change in the geometric phase produces a shift in the fringes under the coherence envelope, but the coherence envelope stays in place, as illustrated in FIG. 4, resulting in no change in the fringe contrast at each point. In comparison, the whole white-light interferogram is shifted during scanning phase shifting, resulting in changing fringe contrast at each point. Thus, for multi-wavelength interferometry a geometric phase-shifter is preferred because it will produce the same phase shifts for any wavelength used in the interferometer.
Achromatic phase-shifters operating on geometric phase have been developed in the art using circularly polarized light, as illustrated in FIGS. 5A and 5B, for example. Linearly polarized light can be achieved by placing first a polarizer in the path of a beam, then passing the light through a quarter-wave plate that makes it circularly polarized, and then through a rotating half-wave plate followed by a quarter-wave plate and another polarizer (FIG. 5A). In a simpler arrangement, the last three elements are substituted by a single rotating polarizer (FIG. 5B). In all cases, in order to introduce a phase shift between two interfering beams, the two beams exiting the interferometer need to be orthogonally polarized (note that then the first polarizer shown in FIG. 5A is not needed).
Based on these principles, FIG. 6 illustrates a Michelson interferometer adapted for geometric phase shifting operation by producing two orthogonally linearly polarized beams at the output of the reference and object arms of the interferometer. A geometric phase-shifter consisting of a rotating half-wave plate mounted between two quarter-wave plates with their axes set at 45° to the angles of polarization of the two beams, is placed at the exit of the interferometer. This interferometer employs the type of geometric phase-shifter shown in FIG. 5A in which the first quarter-wave plate creates left- and right-handed circularly polarized beams. The half-wave plate then changes the right-handed circularly polarized beam to a left-handed one and the left-handed circularly polarized beam to a right-handed one. Finally, the second quarter-wave plate brings the two beams back to their original orthogonal linear polarizations.
As a result of this configuration, a rotation a of the half-wave plate shifts the phase of one linearly polarized beam by +2α and the phase of the other orthogonally polarized beam by −2α, so that a net phase difference of 4α is introduced between the two beams. This phase difference is very nearly independent of the wavelength over the whole visible spectrum. The polarizer makes it possible for the two beams to interfere.
Another type of geometric phase-shifter utilizes the configuration of FIG. 5B, wherein a rotating polarizer is placed after the quarter-wave plate that changed the two orthogonally polarized beams leaving the interferometer to left- and right-circularly polarized beams. In this case, if the test beam is left-circularly polarized and the reference beam is right-circularly polarized and both beams are incident upon the linear polarizer set at an angle α with respect to the x-axis, both the test and reference beams, upon passing through the polarizer, become linearly polarized at an angle α. However, a phase offset +α is added to the test beam and a phase offset −α is added to the reference beam. A rotation of the linear polarizer by α therefore introduces a phase shift 2α between the two interfering beams. The linear polarizer acts as a phase shifting device and also makes it possible for these beams to interfere. While an achromatic quarter-wave plate could be used to extend the spectral range over which this phase-shifter operates, it turns out that the variations in the phase shift produced by this system due to variations in the retardation of the quarter-wave plate with the wavelength are quite small. (See S. S. Helen, M. P. Kothiyal and R. S. Sirohi, “Achromatic Phase-shifting using a Rotating Polarizer,” Opt. Commun., 154, 249-254, 1998).
Because the measurement time can be critical in some industrial applications, it can be reduced significantly if the interferograms are collected simultaneously. This can be done using yet another form of geometric phase-shifting, a pixelated mask, as disclosed in U.S. Pat. No. 7,230,717 (Millerd et al.). As illustrated in FIG. 7, a polarizing beamsplitter is used to produce reference and test beams with orthogonal polarizations. Quarter-wave plates are placed in the reference and test beams so that each beam initially transmitted through the beamsplitter is reflected when it returns, and vice versa. These two beams pass through a quarter-wave plate, which converts the two orthogonally polarized beams to right- and left-handed circularly polarized beams, and then through a phase mask. The quarter-wave plate can be placed at the exit of the interferometer, or in front of the camera, while the phase mask is placed just in front of the CCD array in the camera.
The phase mask is a micropolarizer array built up of groups of four linear polarizer elements having their transmission axes at 0, 45, 90, and −45 degrees (or at 0, 45, −45 and 90 degrees) and is structured so that each polarizer element is placed over a detector element. These four linear polarizer elements introduce phase shifts between the test and reference beams of 0, 90, 180, and 270 degrees. Thus, four phase-shifted interferograms, obtained from each group of pixels, are recorded simultaneously using a single CCD array. As one skilled in the art would easily recognize, the phase mask works as a geometric phase-shifter, the two essential requirements being that the test and reference beams traveling through the quarter wave plate have orthogonal polarizations and that the micropolarizer array match the CCD array.
The solutions described above for effecting geometric phase shifts have been used successfully to overcome the multiple-wavelength and numerical aperture problems discussed above in various interferometer configurations, but no comparable solution has yet been found for Mirau interferometers. In order to use a phase shifter operating on the geometric (Pancharatnam) phase, the Mirau test and reference beams T and R emerging from the interferometer need to be linearly polarized in orthogonal planes. However, because of the compactness of the Mirau interferometric objective and the coaxial configuration of the test and reference arms, very little space is available for introducing additional optical elements and a workable solution has not been found to date in spite of the frequent commercial use of Mirau configurations for optical scanners. The present invention is directed at a viable solution for this problem.